Comments on: On a mixture of the fix-and-relax coordination and Lagrangean substitution schemes for multistage stochastic mixed integer programming

Abstract

It is well known that introducing integer variables into stochastic linear programs with recourse has serious consequences for structural analysis and algorithm design. In particular, convexity of the objective function is lost, preventing, to name just two examples, the direct application of duality reasoning or subgradient based cutting planes. When passing from risk neutral models to formulations allowing for risk aversion, convexity may get lost as well, depending on how risk is quantified. Another source of complication is in passing from two-stage to multistage models. If done in the linear case, this often captures convexity, but it always adds another dimension of complexity, namely dynamics regarding availability of information. The present paper proposes an algorithmic approach to a class of stochastic programs involving all three above mentioned complicating features: integer variables, risk aversion, and multistage decision making. With finite probability spaces, such models allow for equivalent representation as mixed-integer linear programs (MILPs). The bad news, however, is that the sheer dimensionality of the latter usually prevents successful direct application of even the most advanced MILP solution methods and solvers. Rather, the art is in bringing together solution techniques from different branches of integer, or from a broader perspective global, optimization to utilize favorably the hidden structures, mainly decomposition structures. This is the theme of the present paper. The diversity of constraints assembled in a multistage stochastic mixed-integer program, in principle, gives rise to various possible relaxations. In the present paper, the focus is on the integrality and the nonanticipativity constraints. Their relaxation